RGE of a Cold Dark Matter Two-Singlet Model Abdessamad Abadaa,b∗ and Salah Nasrib† a
Laboratoire de Physique des Particules et Physique Statistique,
Ecole Normale Sup´erieure, BP 92 Vieux Kouba, 16050 Alger, Algeria
arXiv:1304.3917v1 [hep-ph] 14 Apr 2013
b
Physics Department, United Arab Emirates University, POB 17551, Al Ain, United Arab Emirates (Dated: April 16, 2013)
Abstract We study via the renormalization group equations at one-loop order the perturbativity and vacuum stability of a two-singlet model of cold dark matter (DM) that consists in extending the Standard Model with two real gauge-singlet scalar fields. We then investigate the regions in the parameter space in which the model is viable. For this, we require the model to reproduce the observed DM relic density abundance, to comply with the measured XENON 100 direct-detection upper bounds, and to be consistent with the RGE perturbativity and vacuum-stability criteria up to 40TeV. For small mixing angle θ between the physical Higgs h and auxiliary field, and DM-h (4)
mutual coupling constant λ0 , we find that the auxiliary-field mass is confined to the interval 116GeV − 138GeV while the DM mass is mainly confined to the region above 57GeV. Increasing (4)
θ enriches the existing viability regions without relocating them, while increasing λ0
shrinks
them with a tiny relocation. We show that the model is consistent with the recent Higgs boson-like discovery by the ATLAS and CMS experiments, while very light dark matter (masses below 5GeV) is ruled out by the same experiments. PACS numbers: 95.35.+d; 98.80.-k; 12.15.-y; 11.30.Qc. Keywords: cold dark matter. light WIMP. extension of Standard Model. RGE.
∗
Electronic address:
[email protected]
†
Electronic address:
[email protected]
1
I.
INTRODUCTION
Now that there is more and more compelling evidence that the discovery in the ATLAS and CMS experiments at the LHC is a Higgs particle with a mass mh ≃ 125GeV [1, 2], one of the main focuses of particle physics is an understanding of the still elusive nature of dark matter (DM), believed to account for about 26% of the energy content of the universe [3]. Side by side with observation, models beyond the Standard Model (SM) are devised to account for weakly interacting massive particles (WIMPs) as plausible candidates for dark matter. Such models range from the sophisticated ones, those bearing intricate underlying symmetries and mechanisms of symmetry breaking, to the more simpler ones, those extending the SM without any particular assumption regarding a deeper structure. Since the findings at the LHC have so far yielded no particular clue as to possible particle structures beyond those of the Standard Model, it is therefore still consistent to try to recognize dark matter as simple WIMPs extending the Standard Model, with no further assumptions as to inner structures. The simplest of such extensions is a one real electroweaksinglet scalar field, interacting with visible matter only via the SM Higgs particle, first proposed by Silveira and Zee [4] and further studied in [5–7]. In this minimal model, with DM masses lighter than 10GeV, the Higgs would be mainly invisible, which is excluded by the recent measurement of the Higgs signal at the LHC [8]. Also, for DM masses in the range 7GeV − 60GeV, the model is ruled out by the data from XENON 10 [9] and CDMS II [10], except around the resonance 62GeV at which the Higgs-DM coupling is extremely small. Furthermore, DM masses between 65GeV and 80GeV are excluded in this one-singlet extension by the XENON 100 experiment [11]. Note that similar conclusions hold also for a complex scalar singlet extension of the SM [12]. So, given the difficulties this minimal model has with existing experimental and observational DM data [13], we have proposed in [14] an extension to the Standard Model with two real electroweak singlets, one being stable, the DM candidate, and the other an auxiliary field with a spontaneously broken Z2 symmetry. Based on the DM relic-density and WIMP direct detection studies, we have concluded that this two-singlet model is capable of bearing dark matter in a large region of the parameter space. Further constraints on the model as well as some of its phenomenological implications have been studied in [15] where rare meson decays and Higgs production channels have been considered. 2
In the present work, we further the study of the two-singlet model and ask how high in the energy scale it is computationally reliable. A standard treatment is the investigation of the running of the coupling constants in terms of the mass scale Λ via the renorrmalization group equations (RGE). We believe one-loop calculations are amply sufficient for the present task; higher loops could be considered if the situation changes. The two standard issues to monitor are the perturbativity of the scalar coupling constants and the vacuum stability of the theory. These issues were studied in [16] for the complex scalar singlet extension of the SM, and it was shown that the vacuum-stability requirement can affect the DM relic density. Specific results from such studies depend on the cutoff scale Λm of the theory. Reversely, imposing perturbativity and vacuum stability may indicate at what Λm the two-singlet extension is valid. In the early parts of this work, the second point of view is adopted, whereas in the later part, the first is taken. Furthermore, up to only recently, it has been anticipated that new physics such as supersymmetry would appear at the LHC at the scale Λ ∼ 1TeV. Present results from ATLAS and CMS indicate no such signs yet. One consequence of this is that the cutoff scale Λm may be higher. As we shall discuss, we find that it can be ∼ 40TeV. This work is organized as follows. After this introduction, we recapitulate in section II the essentials of the two-singlet model necessary for the RGE calculations. In section III, we discuss the running of the scalar coupling constants when we switch off the non-Higgs SM particles. This gives us a first understanding of how high in the energy scale perturbativity is allowed. It helps also seeing the subsequent effects of the other SM particles. Section IV discusses the full RGEs. Vacuum stability gets into focus with the Higgs coupling constant turning negative at some scale. The mass scales at which non-perturbativity and nonstability set in are different, and so a choice for Λm has to be made. Section V attempts at finding the regions in the parameter space in which the model is predictive. In addition to the DM relic-density constraint and the perturbativity-stability criteria deduced from the previous two sections, we impose on the model to be within the current direct-detection experimental bounds. Section VI is devoted to concluding remarks.
3
II.
THE TWO-SINGLET MODEL
The model is obtained by adding to the Standard Model two real, spinless, and Z2 symmetric SM-gauge-singlet fields. One is the dark matter field S0 with unbroken Z2 symmetry, and the other an auxiliary field χ1 with spontaneously broken Z2 symmetry. Both fields interact with the SM particles via the Higgs doublet H. Using the same notation as in [14], the potential function that involves S0 , H and χ1 is: U =
m ˜ 20 2 µ2 S0 − µ2 H † H − 1 χ21 2 2 η0 4 λ η1 λ0 η01 2 2 λ1 † 2 + S0 + H † H + χ41 + S02 H † H + S χ + H Hχ21 , 24 6 24 2 4 0 1 2
(2.1)
where m ˜ 20 , µ2 and µ21 and all the coupling constants are real positive numbers1 .
We are interested in monitoring the running of the scalar coupling constants. A one-loop renormalization-group calculation yields the following β-functions: βη0 = βη1 = βλ = βη01 = βλ0 = βλ1 =
3 16π 2 3 16π 2 3 16π 2 1 16π 2 1 16π 2 1 16π 2
2 η02 + η01 + 4λ20 ;
2 η12 + η01 + 4λ21 ; 3 2 ′2 9 4 4 2 2 2 4 2 2 ′2 λ + λ0 + λ1 − 48λt + 8λλt − 3λg − λg + g g + g ; 3 2 4 2 4η01 + η0 η01 + η1 η01 + 4λ0 λ1 ; 9 3 2 2 2 ′2 ; 4λ0 + λ0 η0 + 2λ0 λ + η01 λ1 + 12λ0 λt − λ0 g − λ0 g 2 2 9 3 2 2 2 ′2 . 4λ1 + λ1 η1 + 2λ1 λ + η01 λ0 + 12λ1 λt − λ1 g − λ1 g 2 2
(2.2)
As usual, βg ≡ dg/d ln Λ where Λ is the running mass scale, starting from Λ0 = 100GeV.
The constants g, g ′ and gs are the SM and strong gauge couplings, known [17] and given to one-loop order by the expression: G (Λ0 ) G (Λ) = r , 2 1 − 2aG G (Λ0 ) ln ΛΛ0 where aG =
−19 , 41 , −7 , 96π 2 96π 2 16π 2
(2.3)
and G (Λ0 ) = 0.65, 0.36, 1.2 for G = g, g ′, gs respectively. The
coupling constant λt is that between the Higgs field and the top quark. To one-loop order, 1
The mutual couplings can be negative as discussed below, see (3.1).
4
it runs according to [17]: βλt with λt (Λ0 ) =
mt (Λ0 ) v
λt = 16π 2
9 2 17 ′2 2 2 , 9λt − 8gs − g − g 4 12
(2.4)
= 0.7, where v is the Higgs vacuum expectation value (vev) and
mt the top mass. Note that we are taking into consideration the fact that the top-quark contribution is dominant over that of the other fermions of the Standard Model. The model undergoes two spontaneous breakings of symmetry: one of the electroweak, with a vev v = 246 GeV, and one of the Z2 symmetry (χ1 field), with a vev v1 we take in this work equal to 150 GeV. Above v, the fields and parameters of the theory are those of (2.1). Below v1 , the (scalar) physical fields are S0 (DM), h (Higgs) and S1 (auxiliary), with parameters (masses and coupling constants) given in Eqs. (2.2–2.15) of [14]. We take the values of the physical parameters at the mass scale Λ0 = 100 GeV. There are originally nine free physical parameters. The two vevs v and v1 are fixed, as well as the mass of the (4)
physical Higgs field mh = 125 GeV [1, 2]. Also, the physical mutual coupling constant η01
between S0 and S1 is determined by the DM relic-density constraint [18], which translates into the condition: hv12 σann i ≃ 1.7 × 10−9 GeV−2 ,
(2.5)
where hv12 σann i is the thermally averaged annihilation cross-section of a pair of two DM particles times their relative speed in the center-of-mass reference frame. This constraint is √ (4) imposed throughout this work, together with the perturbativity restriction 0 ≤ η01 ≤ 4π on its solution. The remaining free parameters of the model are the physical mutual coupling (4)
constant λ0 between h and S0 , the mixing angle θ between h and S1 , the DM mass m0 , the mass m1 of the auxiliary physical field S1 , and the DM self-coupling constant η0 . This latter has so far been decoupled from the other coupling constants [14, 15], but not anymore in view of (2.2) now that the running is the focus. However, its initial value η0 (Λ0 ) is arbitrary and its β-function is always positive. This means η0 (Λ) will only increase as Λ increases, quickly if starting from a rather large initial value, slowly if not. Therefore, without loosing generality in the subsequent discussion, we fix η0 (Λ0 ) = 1. Hence, here too (4)
we still effectively have four free parameters: λ0 , θ, m0 , and m1 . The initial conditions for
5
the coupling constants in (2.1) in terms of these physical free parameters are as follows: 2 3 v 2 2 2 η1 (Λ0 ) = m1 + mh + m1 − mh cos (2θ) + sin (2θ) ; 2v12 2v1 i 2 v1 3 h 2 2 2 m − m cos (2θ) − m + m − sin (2θ) ; λ (Λ0 ) = 1 h 1 h 2v 2 2v sin (2θ) 2 m1 − m2h ; λ1 (Λ0 ) = 2vv1 h i 1 (4) (4) 2 2 η01 (Λ0 ) = η cos θ − λ0 sin θ ; cos (2θ) 01 h i 1 (4) (4) λ0 cos2 θ − η01 sin2 θ . (2.6) λ0 (Λ0 ) = cos (2θ) Note that normally, as we go down the mass scale, we should seam quantities in steps: at v, v1 , and Λ0 . However, the corrections to (2.6) are of one-loop order times ln vv1 or ln Λv10 , small enough for our present purposes to neglect.
III.
SCALARS ONLY
To see the effects of the scalar couplings only and how up in the mass scale the model can go, we switch off the non-Higgs SM couplings in (2.2). The perturbativity constraint √ we impose on all dimensionless scalar coupling constants is G (Λ) ≤ 4π. Vacuum stability means that G (Λ) ≥ 0 for the self-coupling constants η0 , λ, and η1 , and the conditions: −
√ 1p η0 λ ≤ λ0 ≤ 4π; 6
−
√ 1√ η0 η1 ≤ η01 ≤ 4π; 6
−
√ 1p η1 λ ≤ λ1 ≤ 4π 6
(3.1)
for the mutual couplings λ0 , η01 , and λ1 . Also, as a start, we let the masses m0 and m1 vary in the interval 1GeV − 200GeV.
Fig. 1 displays a typical running of the scalar self-coupling constants, from Λ0 = 102 GeV
up to 1012 GeV. As is expected for scalars only, all coupling constants are increasing functions of the scale Λ, with different but increasing rates. Also, the larger value the coupling starts from at Λ0 , the faster it will go up. Fig. 2 shows the running of the mutual coupling constants for the same values of the parameters. For these values, the mutual coupling constants start well below 1, and so run low; they are very much dominated by the self-couplings. This situation will stay for λ0 and λ1 in all regions, but not for η01 . The first coupling constant that leaves the perturbativity bound
√
4π is η1 , the self-
coupling constant of the auxiliary scalar field χ1 , at about 1260 TeV for this set of values of 6
Λ0 H4L = 0.01, Θ = 10° , m0 = 55GeV, m1 = 110GeV
8 g = Η0
6
g = Η1
gHTL
g=Λ
4
2
0 0
2
4
6
8
10
T = Log10 HLL0 L FIG. 1: The running of the self-couplings (scalars only). The self-coupling η1 of the auxiliary field χ1 dominates over the Higgs self-coupling λ.
the parameters. This behavior is in fact typical. Indeed, η1 starts above 2 at Λ0 in all the parameter space, much higher than all the other coupling constants – only η01 can compete with it in some regions. As it intervenes squared in its own β-function, it will also move up quicker. More precisely, from (2.6), we see that η1 (Λ0 ) depends on m1 and θ only. The effect of the mixing angle θ is small. As a function of m1 , starting from about 2, η1 (Λ0 ) √ decreases slightly until mh and then picks up. It will pass the perturbativity bound 4π at about m1 ≃ 160GeV. This means that the region m1 > 160GeV is automatically excluded from the outset. In actual situations, given the positive-slope RG running of η1 (Λ) and even in the case of the full RGE (see below), perturbativity puts a stricter upper bound on m1 , irrespective of the other parameters of the model. In Fig. 1, the Higgs self-coupling λ starts just above 0.6 and does not pick up much when running. This behavior is typical too. Indeed, λ (Λ0 ) is also a function of m1 and θ only, see (2.6). For a given θ, it will increase as a function of m1 to reach 3 (mh /v)2 at m1 = mh , equal here to 0.77 for mh = 125GeV. Then it continues to increase, but with a smaller
7
Λ0 H4L = 0.01, Θ = 10° , m0 = 55GeV, m1 = 110GeV
0.05 g = Η01
0.04 g = Λ0 g = Λ1
gHTL
0.03
0.02
0.01
0.00 0
2
4
6
8
10
T = Log10 HLL0 L FIG. 2: Running of the mutual couplings (scalars only). For these values of the parameters, all three are well below the self-couplings.
slope. The mixing angle θ enhances the behavior of λ (Λ0 ) as a function of m1 , but for, say2 θ = 15o , λ (Λ0 ) will be less than 0.85 at m1 = 160 GeV. This situation implies that when running as a function of the scale Λ, the Higgs self-coupling λ will increase, but will hardly reach 1 before, say η1 , leaves the perturbativity bound. Increasing the dark-matter mass does affect the running of the couplings. Figs. 3 (selfcouplings) and 4 (mutual couplings) display such effects for m0 = 100GeV. Among the self-couplings, η1 is still dominant, but tailed more closely by η0 this time. For both, the positive acceleration is accentuated, something that makes η1 leave the perturbativity region much earlier, at about 6.3TeV. By contrast, the running of the Higgs self-coupling λ stays flat. The major effect of increasing m0 is on the mutual coupling η01 , between the DM field S0 and the auxiliary field χ1 . Indeed, in Fig. 2 where m0 = 55GeV, η01 started and ran small 2
We are implicitly confining the mixing angle θ to small values, a situation inferred from our work [15] on the phenomenological implications of the model. This is discussed later in section V.
8
Λ0 H4L = 0.01, Θ = 10° , m0 = 100GeV, m1 = 110GeV
g = Η0
15
g = Η1
gHTL
g=Λ
10
5
0 0
1
2
3
4
T = Log10 HLL0 L FIG. 3: Running of the self-couplings (scalars only) for a larger DM mass. η1 is still dominant, even if η0 picks up faster behind it.
like the other two mutual couplings. Here, whereas λ0 and λ1 (both Higgs related) stay close to zero, η01 starts above 2.5 and runs up fast. In fact, for these values of the parameters, it leaves the perturbativity region earlier than η1 , at about 2.5TeV. Increasing the auxiliary-field mass m1 has a similar effect: it enhances the positive acceleration of the self-couplings η1 and η0 while leaving λ flat, and boosts up the mutual coupling η01 away from λ0 and λ1 , which both remain not far from zero. It also makes η1 and η01 leave the perturbativity region earlier, without η01 necessarily taking over from η1 . (4)
Increasing λ0 (4)
λ0
has also an effect. Figs. 5 (self) and 6 (mutual) show the running for
= 0.4. The self-coupling η1 dominates and leaves the perturbativity region at about
251TeV. The mutual coupling η01 is raised above 1 at Λ0 and so can run high, while λ0 and λ1 stay here too just above zero. It leaves the perturbativity region at about 794GeV, well (4)
behind η1 . Higher values of λ0 are more difficult to achieve as the relic-density constraint (2.5) may not be satisfied [14]. Finally, changing the mixing angle θ has little effect on the self-coupling constants. It
9
Λ0 H4L = 0.01, Θ = 10° , m0 = 100GeV, m1 = 110GeV
12 g = Η01
10
g = Λ0
8 gHTL
g = Λ1
6 4 2 0 0
1
2
3
4
T = Log10 HLL0 L FIG. 4: Running of the mutual couplings (scalars only) for a larger DM mass. η01 starts above 1, much higher than the two others, even higher than the self-coupling η1 .
helps the mutual coupling constants η01 and λ1 start higher, but not by much: they stay with λ0 well below one.
IV.
THE FULL RGE
In the previous situation, ‘scalars only’, all running coupling constants were positive, and so there were no issues related to vacuum stability. We now reintroduce the other SM particles and see their effects. Fig. 7 displays the behavior of the self-couplings under the full RGE for the same values of the parameters as in Fig. 1 (scalars only). The dramatic effect is on the Higgs self-coupling constant λ which quickly gets into negative territory, at about 15TeV, thus rendering the theory unstable beyond this mass scale. This is better displayed in Fig. 8 where the RG behavior of λ is shown by itself. Such a negative slope for λ is expected, given the negative contributions to βλ in (2.2). Here too η1 is dominant over the other couplings and still controls perturbativity, leaving its region much later, at about 1600 TeV, farther away from the situation ‘scalars only’. This looks to be a somewhat general 10
Λ0 H4L = 0.4, Θ = 10° , m0 = 50GeV, m1 = 110GeV
14 g = Η0
12
g = Η1
10
gHTL
g=Λ
8 6 4 2 0 0
1
2
3
4
5
6
T = Log10 HLL0 L (4)
FIG. 5: Running of the self-couplings (scalars only) with a larger physical Higgs self-coupling λ0 . The self-coupling η1 is still dominant.
trend: the non-Higgs SM particles seem to flatten the runnings of the scalar couplings. The runnings of the mutual coupling constants for the same set of parameters’ values is displayed in Fig. 9. They also get flattened by the other SM particles, but they stay positive. Here too they dwell well below the self-couplings, as in the ‘scalars only’ case. In fact, many of the effects on the running coupling constants coming from varying the parameters are similar to those of the previous situation since the SM particles do not intervene in the initial values of the couplings (self and mutual) at Λ0 . This means that increasing m0 and m1 will raise the mutual coupling η01 and not the two others, higher than η1 in some regions. For example, Fig. 10 shows the running of the self-couplings when m0 = 100GeV. Both η1 and η0 run faster but λ is little affected. Fig. 11 shows the running of the mutual couplings from the full RGE also at m0 = 100GeV. As in the case ‘scalars only’, larger m0 boosts up η01 (Λ0 ), much higher than λ0 and λ1 , at about 2.2 here, which makes it run quickly high, leaving the perturbativity region before η1 , as in the case ‘scalars only’. (4)
Raising λ0
will also make the self-couplings η1 and η0 run faster while affecting very
11
Λ0 H4L = 0.4, Θ = 10° , m0 = 50GeV, m1 = 110GeV
7 6
g = Η01
5
g = Λ0 g = Λ1
gHTL
4 3 2 1 0 0
1
2
3
4
5
6
T = Log10 HLL0 L (4)
FIG. 6: Running of mutual couplings (scalars only). Larger λ0
helps η01 rise well above λ0 and
λ1 , but not enough to win over the self-coupling η1 .
little λ. It will also make the mutual coupling η01 starts higher, and so demarked from λ0 and λ1 . By contrast, the effect of θ is not very dramatic: the self-couplings are not much affected and the mutuals only evolve differently, without any particular boosting of η01 .
V.
REGIONS OF VIABILITY
The foregoing discussion showed us how the scalar parameters of the two-singlet model behave as a function of the mass scale Λ. From the situation ‘scalars only’ we understood that the two couplings that control perturbativity are η1 and η01 . The full RGE brought in stability: the change of sign of λ is the vacuum stability criterion to use. Equipped with these indicators, we can try to investigate in a more systematic way the viability regions of the model, regions in the space of parameters in which the model is predictive. Remember that this model has four parameters: the dark-matter mass m0 , the physical auxiliary field (4)
mass m1 , the physical Higgs self-coupling λ0 , and the mixing angle θ between the physical (4)
Higgs and the auxiliary field. The way we proceed is to vary λ0 and θ and try to find the 12
Λ0 H4L = 0.01, Θ = 10° , m0 = 55GeV, m1 = 110GeV
10 g = Η0
8 g = Η1
gHTL
6
g=Λ
4
2
0 0
2
4
6
8
10
T = Log10 HLL0 L FIG. 7: Running of the self-couplings (full RGE). η1 controls perturbativity and the Higgs coupling λ becomes negative quickly.
regions of viability of the model in the (m0 , m1 )-plane. We have by now a number of tools at our disposal. First the DM relic-density constraint (2.5), which has been applied throughout and will continue so. We have the RGE analysis √ of this work. We will require both η1 (Λ) and η01 (Λ) to be smaller than 4π, and λ (Λ) to be positive. There is one important issue to address though before we proceed, and that is how far we want the model to be perturbatively predictive and stable. The maximum value Λm for the mass scale Λ should not be very high for two reasons. One, more conceptual, is that we want to recognize and allow the model to be intermediary between the Standard Model and some possible higher structure. The second reason, more practical, is that a too-high Λm is too restrictive for the parameters themselves. For example, for the parameters we used in the previous sections, in particular m0 = 55GeV and m1 = 110 GeV, we have seen that λ gets negative already for Λ ≃ 15TeV whereas η1 leaves the perturbativity region much later, for Λ ≃ 1600TeV. The situation can be reversed. For example, for m0 = 67GeV, 13
Λ0 H4L = 0.01, Θ = 10° , m0 = 55GeV, m1 = 110GeV
0.6
0.4
ΛHTL
0.2
0.0
-0.2
-0.4
-0.6 0
2
4
6
8
10
T = Log10 HLL0 L
FIG. 8: The running of the Higgs self-coupling λ (full RGE). It gets negative at about 15TeV for this set of parameters’ values.
m1 = 135GeV, and θ = 15o , λ can live positive until about 400TeV whereas η1 leaves perturbativity at about 50TeV. In this section, we set Λm ≃ 40TeV. As a third viability tool, we want the model to comply with the measured direct-detection upper bounds. In our model, the total cross section for non-relativistic elastic scattering of a dark matter WIMP off a nucleon target is given by the relation [14]: #2 2 " (3) (3) m2N mN − 97 mB λ0 cos θ η01 sin θ σdet = . − m2h m21 4π (mN + m0 )2 v 2
(5.1)
In this relation, mN is the nucleon mass and mB the baryon mass in the chiral limit. The (3)
(3)
quantities λ0 and η01 are coupling constants of cubic terms in the theory after spontaneous breaking of the two symmetries [14]: (3)
λ0 = λ0 cos θ + η01 v1 sin θ;
(3)
η01 = η01 v1 cos θ − λ0 v sin θ.
(5.2)
The condition we impose is that σdet be within the XENON 100 upper-bounds [11]. In work [15], we studied phenomenological implications of the model and constraints on it, using rare meson decays and Higgs production. A number of inferences were deduced, but 14
Λ0 H4L = 0.01, Θ = 10° , m0 = 55GeV, m1 = 110GeV
0.04
g = Η01 g = Λ0
0.03 gHTL
g = Λ1
0.02
0.01
0.00 0
2
4
6
8
10
T = Log10 HLL0 L FIG. 9: Running of the mutual couplings (full RGE). The inclusion of the other SM particles flattens the runnings.
we will prefer to retain only two. One is that the mixing angle θ is to be chosen small. This is emphasized in view of the mounting evidence of a SM Higgs particle found by ATLAS (4)
and CMS at the LHC [1, 2]. The other is that the physical self-coupling λ0 is to be small too. This was already observed in [14], where the relic-density constraint has the tendency (4)
of ‘shutting down’ high values of λ0 . At the end of the next section, we will comment on (4)
possible larger values for λ0 . In this section, the display range of m0 and m1 is from 1GeV to 160GeV. Indeed, there is no reliable data to discuss regarding a dark-matter mass below the GeV, and in view of the behavior of η1 at Λ0 as a function of m1 , taking this latter beyond 160GeV is outside the perturbativity region. In practice, m0 was taken up to 200GeV, with no additional features to report. (4)
Let us start with λ0 and θ both very small. Fig. 12 displays the regions (blue) for which (4)
the model is viable up to Λm ≃ 40TeV. Here λ0
= 0.01 and θ = 1o . We see that the
mass m1 is confined to the interval 116GeV − 138GeV. The dark-matter mass is confined 15
Λ0 H4L = 0.01, Θ = 10° , m0 = 100GeV, m1 = 110GeV
g = Η0
15 g = Η1
gHTL
g=Λ
10
5
0 0
1
2
3
4
T = Log10 HLL0 L FIG. 10: Running of the self-couplings (full RGE) with m0 larger. η1 is more closely tailed by η0 and λ decreases and turns negative at about 10TeV.
mainly to the region above 118GeV, the left boundary of which having a positive slope as m1 increases. The DM mass m0 has also a small showing in the narrow interval 57GeV−68GeV. The effect of increasing the mixing angle θ is to enrich the existing regions without relocating them. This is displayed in Figs. 13 and 14 for which θ is increased to 5o and 15o respectively. We see that, as θ increases, the region between the narrow band and the larger one to the right gets populated. This means more dark-matter masses above 60GeV are allowed, but m1 stays in the same interval, roughly 116GeV − 138GeV. (4)
Increasing the Higgs-DM mutual coupling λ0
has the opposite effect, that of shrink(4)
ing existing viability regions. Indeed, compare Fig. 15 for which λ0
= 0.1 and θ = 15o
(Λm ≃ 40TeV) with Fig. 14. We see shrunk regions, pushed downward by a few GeVs, (4)
which is not a substantial relocation. Remember that increasing λ0
raises η01 (Λ0 ) well
enough above 1 so that this latter will leave the perturbativity region sooner. Increasing it is also caught up by the relic-density constraint, which tends to shut down such larger (4)
values of λ0 when the dark-matter mass m0 is large. The direct-detection constraint has
16
Λ0 H4L = 0.01, Θ = 10° , m0 = 100GeV, m1 = 110GeV
12 g = Η01
10
g = Λ0
8 gHTL
g = Λ1
6 4 2 0 0
1
2
3
4
T = Log10 HLL0 L FIG. 11: Running of the mutual couplings (full RGE) with m0 larger. η01 starts well above λ0 and λ1 and leaves the perturbativity region before the self-coupling η1 .
also a similar effect.
VI.
CONCLUDING REMARKS
In this work, we have studied the effects and consequences of the renormalization group equations at one-loop order on a two-singlet model of cold dark matter that consists in extending the Standard Model with two real gauge-singlet scalar fields. The two issues we monitored are perturbativity and vacuum stability. The former is controlled by the auxiliary-field self-coupling η1 and the mutual coupling η01 between the dark matter and the auxiliary fields. The latter is controlled by the Higgs self-coupling λ. When the non-Higgs SM coupling constants are switched off, all scalar couplings are positive increasing functions of the scale Λ. Reintroducing them flattens the rates for all the scalar couplings and makes the Higgs coupling λ turn negative at some scale. The mutual couplings λ0 (DM-Higgs) and λ1 (Higgs-auxiliary) stay always well below one, whereas η01 boosts up for larger m0 (DM mass) and/or m1 (auxiliary-field mass), dominating over η1 in some regions. 17
Λ0 H4L = 0.01, Θ = 1°
0
50
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m1 HGeVL
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m0 HGeVL (4)
FIG. 12: Regions of viability of the two-singlet model (in blue). Physical Higgs self-coupling λ0 and mixing angle θ very small.
We then have investigated the regions in the space of parameters in which the model is viable. We have plotted these regions in the (m0 , m1 )-plane while varying the physical mutual (4)
coupling λ0 between the dark matter S0 and the physical Higgs h, and the mixing angle θ between h and the physical auxiliary field. We have required that the model reproduces the DM relic density abundance, and that it complies with the measured direct-detection upper bounds – those of the XENON 100 experiment. We have also imposed the RGE perturbativity and vacuum-stability criteria that we deduced from this work together with a maximum cutoff Λm ≃ 40TeV, a scale at which heavy degrees of freedom may start to be relevant, something that could be probed by future colliders. This analysis has shown that (4)
for small λ0 and θ, the auxiliary-field mass m1 is confined to the interval 116GeV−138GeV, while the DM mass m0 is confined mainly to the region above 118GeV, with a small showing 18
Λ0 H4L = 0.01, Θ = 5°
0
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FIG. 13: Regions of viability (blue) of the model. λ0 still very small, but θ larger. The region is richer, but not relocated.
in the narrow interval 57GeV − 68GeV. Increasing θ enriches the existing viability regions (4)
without relocating them, while increasing λ0 has the opposite effect, that of shrinking them without substantial relocation. It is pertinent at this stage to comment on the implications of the Higgs discovery at the LHC on the possibility of having a light dark matter WIMP S0 with a mass m0 . 62GeV, a situation allowed in this two-singlet model. Indeed, on the one hand, for such a light dark matter, the decay channel h → S0 S0 becomes open, and therefore will lower the number of Higgs decays into SM particles. On the other hand, The ATLAS and CMS published data on Higgs boson searches seem to indicate that the observed boson is SM-like, and so, one expects to have stringent constraints on the parameter space when it comes to light darkmatter masses. In [19], a global fit to the Higgs boson data that includes those presented 19
Λ0 H4L = 0.01, Θ = 15°
0
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FIG. 14: The region of viability (blue) is even richer for larger mixing angle θ.
at the Moriond 2013 conference by the ATLAS and CMS collaborations [20, 21] has been performed; see [22] for earlier fits. It has been found that any extra invisible Higgs boson decay must be bounded by the following condition on the corresponding branching ratio: Br(h → invisible) < 19%.
(6.1)
It turns out that in our two-singlet model, the branching fraction of the invisible width of the Higgs boson is smaller than the bound above for m0 . 62GeV. Indeed, if we take for example m0 = 55GeV used frequently in this work, the ratio Γ(h → S0 S0 )/Γ(h → b¯b) is less than 17%, quite consistent with the above current bound. Therefore, we conclude that the two-singlet model is consistent with the current available data regarding the Higgs boson searches. Finally, we ask whether the model allows for very light cold dark matter. Below 5GeV, 20
Λ0 H4L = 0.1, Θ = 15°
0
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m0 HGeVL (4)
FIG. 15: The physical Higgs self-coupling λ0 shrinks the viability region (blue) as it increases.
direct detection puts no experimental bound on the total cross section σdet for non-relativistic elastic scattering of a dark matter WIMP off a nucleon target. Such a situation allows for (4)
very small m0 regions of viability, but only when λ0 is quite large (∼ 2 and above) and θ not too small ( ∼ 15o and above). However, for such values of the parameters, the branching fraction of the invisible Higgs decay is larger than 25%, which is excluded by the current LHC available data.
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